Stresa, Italy, 25-27 April 2007
EXPERIMENTAL CHARACTERIZATION OF THE STATIC BEHAVIOUR OF
MICROCANTILEVERS ELECTROSTATICALLY ACTUATED
Alberto Ballestra1, Eugenio Brusa2, Mircea Georghe Munteanu2, Aurelio Som?1
1 Laboratory of Microsystems, Department of Mechanics, Politecnico di Torino,
C.so Duca degli Abruzzi, 24 ? 10129 Torino, Italy; alberto.ballestra@polito.it, aurelio.soma@polito.it
2 Department of Electrical, Management and Mechanical Engineering, Universit? di Udine,
via delle Scienze, 208 ? 33100 Udine, Italy; eugenio.brusa@uniud.it, munteanu@uniud.it
ABSTRACT
This paper concerns the experimental validation of some
mathematical models previously developed by the
authors, to predict the static behaviour of
microelectrostatic actuators, basically free-clamped
microbeams. This layout is currently used in RF-MEMS
design operation or even in material testing at microscale.
The analysis investigates preliminarily the static
behaviour of a set of microcantilevers bending in-plane.
This investigation is aimed to distinguish the geometrical
linear behaviour, exhibited under small displacement
assumption, from the geometrical nonlinearity, caused by
large deflection. The applied electromechanical force,
which nonlinearly depends on displacement, charge and
voltage, is predicted by a coupled-field approach, based
on numerical methods and herewith experimentally
validated, by means of a Fogale Zoomsurf 3D. Model
performance is evaluated on pull-in prediction and on the
curve displacement vs. voltage. In fact, FEM nonlinear
solution performed by a coupled-field approach, available
on commercial codes, and by a FEM non-incremental
approach are compared with linear solution, for different
values of the design parameters.
1. INTRODUCTION
In microsystem mechanical design cantilever beams are
currently widely used, as basic components in
microsensors, microswitches and RF-MEMS as well as in
experimental micromechanics, whose goal is
characterizing the materials mechanical properties and
strength, at microscale [1,2,3,4]. The latter aspects
motivate the implementation of efficient numerical models
to predict the electromechanical behaviours of such
microdevices, under the actuation of the electric field, as
stand-alone systems or better as structural components of
assembled parts, as recent DTIP Conferences showed
during the last years, like in [5-8]. Model validation is
currently performed not only to verify the effectiveness of
proposed analytical, numerical and even compact
approaches [9, 10], but to define the model sensitivity on
the uncertainties about the actual values of the design
parameters and of materials properties, whose
measurement is often fairly difficult. A couple of targets
appear currently challenging for structural
micromechatronics. An assessment of accurate coupled-
field models and numerical solutions shall allow a
coherent interpretation of the specimen response in all the
experimental procedures, currently performed and aimed
to characterize both the materials and the MEMS layouts
[3, 4, 11, 12, 13]. Moreover, to build effective numerical
simulators, able to predict the coupled behaviour of
MEMS within the whole electronic circuit, only a
validation of each single model included in a hierarchical
approach will allow satisfying the requirement [9]. This
paper contributes to the above mentioned tasks, by
investigating the effectiveness and the computational
performance of the numerical models proposed in [1, 14-
18], dealing with the static behaviour of microcantilevers.
Moreover, the above mentioned models have to be even
used in dynamic analysis algorithms, when geometrical
nonlinearity has to be added to the effects of nonlinear
electromechanical [16-18].
2. THE EXPERIMENTAL SET-UP
2.1. Microcantilevers with in-plane bending
A first group of specimens including free-clamped
microcantilevers was designed and built, according to the
design rules and the process constraints imposed by
microfabrication, followed by STMicroelectronics
(Cornaredo, Italy). Process ?Thelma? allows a gradual
growth of thick polysilicon layers, being suitable to
fabricate cantilever beams, for which the bending
deflection occurs in-plane, with respect to the reference
plane of the wafer (Fig.1). This approach was followed to
validate the developed models, by means of the
experimental measures performed by Fogale Zoomsurf 3D
[19]. All microspecimens consist of a massive electrode
?EDA Publishing/DTIP 2007
ISBN: 978-2-35500-000-3
A.Ballestra, E.Brusa, M.Gh. Munteanu, A.Som?
Experimental characterization of the static behavior of microcantilevers electrostatically actuated
where is clamped a thin microbeam, bending across the
gap towards a massive counter-electrode.
Microbeam
Supporting
frame
Counter-
electrode
Figure 1: Microcantilever specimen in-plane bending
Microcantilever is a part of a wider structure, equipped by
a connection pad. Thickness and length are measured in
the plane of the wafer, while the width is measured along
the direction orthogonal to the wafer plane. The electric
potential is imposed on the electrode, where microbeam is
clamped, through the pad and to the connection pad of the
counter-electrode, thus applying the voltage through the
gap. To perform a parametric investigation several lengths
were foreseen, as well as different values of gaps were
obtained. Dimensions are listed in Table 1. The material
of specimens is PoCl3 doped epitaxial polysilicon, with
E = 166000 MPa and v =0,23.
Width was imposed by the microfabrication process,
being the thickness of the epitaxial polysilicon layer, as
well as the distance between the microbeam and the
silicon layer underneath located, being 4.1 ?m,
corresponds to the thickness of the sacrificial silicon oxide
layer removed by etching. The massive structure of the
electrode supporting the microbeam helped
microfabrication process to obtain the longest beams by
etching. In the electromechanical coupling it regularizes
the electrostatic field across the gap. An optimised value
of 2 ?m of thickness was found as compromise between
the need of a sufficient electrostatic actuation to bend the
specimen and the electric breakdown. Length, width and
gap values were selected to have a good variety of aspect
ratios, as it is discussed in following paragraphs.
Connection pads, obtained by deposition of Aluminium
alloy layer, offer a square contact area, whose side is 80
?m, to allow a stable contact to the probes used to apply
the electrostatic actuation (Fig.2). A third pad, located on
the edge of the die is connected to the silicon layer under
the beam. Two connection pads have been placed on the
counter-electrode, in order to keep the probes distant
enough to avoid parasitic effects and mutual interference.
950 ?m
800 ?m
cantilever beam
connection pads
gap
counter-electrode
Figure 2: Example of microcantilever.
Geometrical dimensions of the microbeams were
measured by means of Fogale Zoomsurf 3D. Some
differences between the nominal and the actual values,
due to the process tolerances, were detected, as shown in
Table 1. Listed values include a range of variation of the
parameters on the population of specimens, having equal
geometry, up to seven microstructures. Experiments were
performed by identifying the single specimen within the
group having the same reference code, in Table 1. Code
ST1 identifies this first set of specimens, while each
layout is identified by the second number, like ST1-1.
Measured length l, width w, thickness t, gap g and aspect
ratios from R1 to R4 are evidenced by symbol (*).
2.2. Remarks on the aspect ratios of microspecimens
Specimens were designed by taking care of four basic
aspect ratios, which affect their mechanical behaviour:
lwR /1 = ; lgR /2 = ; ltR /3 = ; wtR /4 = (1)
In particular, R1 may warn about the limit of application
of beam model with respect to the plate?s one [1]; R2
foresees the possibility of large displacement [1], i.e.
geometrical nonlinearity [14-18], while R3 and R4 are used
to evaluate the beam stiffness, even to predict the
anticlastic curvature [1]. Results show that specimens
ST1-1,ST1-2, ST1-3 may need to resort to plate model.
Width values may motivate a certain influence of the three
dimensional nature of the electric field, affecting the
actual value of the electromechanical force. The lateral
curvature does not seem dominant, to require to include
this deformation in the models. Specimens ST1-3, ST1-7,
ST1-8 are prone to exhibit geometrical nonlinearity,
caused by large tip displacement, if compared to the
length of the beam. Fairly compliant are microbeams ST1-
6;-7;-8.
2.3. Experimental set-up
Experimental validation was performed by the optical
profiling system Fogale Zoomsurf 3D, based on non-
contact optical interferometry [19]. The maximum lateral
resolution is similar to that of the conventional optical
?EDA Publishing/DTIP 2007
ISBN: 978-2-35500-000-3
A.Ballestra, E.Brusa, M.Gh. Munteanu, A.Som?
Experimental characterization of the static behavior of microcantilevers electrostatically actuated
microscopes (diffraction limited, 0.6?m with a 20X
objective), while the vertical resolution may reach 0.1 nm.
Optical magnification can reach up to 32X. The recorded
light intensity is detected by a CCD pixel as function of
the specimen height, thus defining either the profile of the
monitored specimen or its position within the lighted area.
Microbeams were fabricated on square chips of 3 mm. To
prevent any accidental motion of the chip, the latter was
fixed on the motorized XY in-plane translation stage of the
profiler, by a glass slide. The objective is equipped by a
motorized Z translation stage, allowing the motion along
the column, which is controlled.
ID Ner l w t g R1 R2 R3 R4 w* R1* R2* R3* R4*
ST1-1 100 15 2 5 0,150 0,050 0,020 0,133 101,00 ?0,1 15,00 1,80 ?0,02 5,00 ?0,3 0,149 0,050 0,018 0,120
ST1-2 100 15 2 10 0,150 0,100 0,020 0,133 101,00 ?0,1 15,00 1,80 ?0,02 10,00 ?0,3 0,149 0,099 0,018 0,120
ST1-3 100 15 2 20 0,150 0,200 0,020 0,133 101,00 ?0,1 15,00 1,80 ?0,02 20,10 ?0,3 0,149 0,199 0,018 0,120
ST1-4 200 15 2 10 0,075 0,050 0,010 0,133 205,00 ?0,2 15,00 1,90 ?0,02 10,00 ?0,3 0,073 0,049 0,009 0,127
ST1-5 200 15 2 20 0,075 0,100 0,010 0,133 205,00 ?0,2 15,00 1,90 ?0,02 20,00 ?0,3 0,073 0,098 0,009 0,127
ST1-6 800 15 2 40 0,019 0,050 0,003 0,133 805,00 ?0,5 15,00 2,70 ?0,04 39,60 ?0,3 0,019 0,049 0,003 0,180
ST1-7 800 15 2 200 0,019 0,250 0,003 0,133 805,00 ?0,5 15,00 2,70 ?0,04 200,00 ?0,5 0,019 0,248 0,003 0,180
ST1-8 800 15 2 400 0,019 0,500 0,003 0,133 805,00 ?0,5 15,00 2,70 ?0,04 400,00 ?0,5 0,019 0,497 0,003 0,180
l* t* g*
Table 1:Synoptic table of nominal and actual dimensions (*) and related aspect ratios
of set ST1 of in-plane bending microbeams (units [?m]).
Microbeam is bended by the electromechanical action,
induced by the electric field, when voltage is applied
between the beam electrode and the counter-electrode,
through the connecting pads. Power supplier is internal in
Fogale Zoomsurf 3D and supplies only up to 200 Volt.
Connection between power supplier and circuit was
assured by adjustable needles, mounted on the
ProbeHeads PH100 Suss. The latter have a mobile arm,
with a pivot, which was magnetically fixed on the work
plane of the instrument (Fig.3). The needle position was
driven on the pad by means of three screws, controlling
the motion along the three directions. Tests were
performed by applying a positive voltage to the counter-
electrode and connecting the beam, the electrode and the
silicon wafer all together to the ground (null voltage).
This configuration avoids unforeseen deflections of the
microbeam under the bias voltage and minimizes the
fringing field effect. Static deflection was detected by
processing the high resolution images, obtained by white
light measurement, through a scanning of the
interferometric fringes on the focused area of the
monitored specimen. In practice, scanning rectangle
included the tip of the beam and part of the massive
element of the electrode, as in Figure 4. Interferometric
measurement provided a top view of the specimen, limited
to the focused window, then a quoted profile of the
transversal section of the microbeam (Fig.4).
3. THE EXPERIMENTALVALIDATION
Models validation was based on the experimental
reconstruction of the curve displacement vs. voltage, to
verify, point by point, the correspondence of the actual tip
displacement, measured by Zoomsurf 3D and the
predicted numerical values. A geometrical linear solution,
which assumes small displacement, is compared to the
nonlinear approaches, implemented by means of the Finite
Element Method (FEM).
Figure 3: Experimental set-up on Fogale Zoomsurf 3D.
The electromechanical coupling between the electrical
and mechanical degrees of freedom motivates to resort to
a so-called coupled-field analysis, including both
mechanical and electrical degrees of freedom. The
original method proposed by the authors in [16-18],
consists of a non incremental solution of the coupled
problem, made possible by introducing a special finite
beam element (so called SFET) suitable to operate even in
presence of large displacement. This method is compared
to the results of a coupled-field approach, based on a
FEM iterative solution, which applies a morphing of the
elements in the dielectric region, to avoid the effects of
the element distortions. This approach is available in
commercial code ANSYS, by meshing elements
PLANE121 and PLANE183. All numerical outputs were
compared to the experimental results in Figures 5, 6, 7.
Plane models were initially implemented, to distinguish
the effects of nonlinear electromechanical coupling, from
those due to the three dimensional nature of the problem.
A complete investigation about the differences between
two and three dimensional models are currently carried
?EDA Publishing/DTIP 2007
ISBN: 978-2-35500-000-3
A.Ballestra, E.Brusa, M.Gh. Munteanu, A.Som?
Experimental characterization of the static behavior of microcantilevers electrostatically actuated
out and validated. In practice, these investigations
demonstrate the presence of local effects, in the electric
field distribution, affecting the actual value of the
electrostatic forces and somewhere the pull-in prediction.
Figure 4: Experimental images and profile provided by the Zoomsurf 3D Fogale.
Results in terms of pull-in voltage were compared even to
the analytical simplified solutions, computed by means of
lumped parameters models proposed in [21]. For each set
of specimens the validation has been completed, by
experimentally measuring the tip displacement of the
microcantilever, for a gradually increasing voltage, up to
pull-in, every time it was allowed by the operating
conditions. Moreover, the same measurement was
repeated several times and averaged on the same sample,
up to twelve times, depending on the occurrence of
accidental failure or destructive pull-in.
3.1. Analytical approaches
A preliminary analysis was performed on pull-in voltage
and related displacement to give a figure of the expected
values on the experiments, by means of the well known
formulas proposed in [21]. Results are immediately
compared to the experimental evidences, where it was
possible, in Table 2. Since several specimens exhibit pull-
in voltage above the limit of 200 V of the Zoomsurf 3D
power supplier, in absence of an external supplier,
comparison was limited to the maximum value of voltage
reached. Prediction of pull-in parameters looks quite
good, although approximated, if performed according to
Senturia-Osterberg formulation as [1, 21]:
2
32
0
4
PI
04
33
0
PI 4
3;
42.01
28.0
PIVtEg
lv
w
gl
EtgV ?
?
=
?????? +
= (2)
where symbols mean: g0 initial gap, t thickness, E
Young?s modulus, ? dielectric constant, l length, VPI pull-
in voltage, vPI pull-in displacement.
3.2. FEM approaches
A complete experimental validation was performed on the
set of specimens ST1, described in Table 1. The most
relevant results are herewith summarized and compared in
figures 5-7.
A first nonlinear and non incremental approach, suitable
to predict even large displacement was implemented in
MATLAB, according to [16-18]. FEM discretization was
applied to consider only the most significant specimens
ST1-1/4/6, mesh was generated as follows. The structure
was described by 20 3-node SFET elements (Special
beam element) [17, 18] for a total of 41 nodes, and for the
dielectric 5633 nodes, with 2672 6-node isoparametric
triangular finite elements for ST1-1, 5409 nodes and
2544 6-node isoparametric triangular finite elements for
ST1-4 and 5301 nodes, 2472 6-node isoparametric
triangular finite elements ST1-6. All the above models
were implemented by the authors in MATLAB.
The latter method was compared to the iterative approach,
including mesh morphing and geometrical nonlinear
solution, implemented for instance in ANSYS, through
PLANE183, 8-node isoparametric quadrilateral finite
elements, solid (beam) and PLANE 121 8-node
isoparametric quadrilateral finite elements (electrostatic).
In this case a suitable mesh consisted of 80 beam elements
and started with 9500 nodes and 3000 elements, but the
number of PLANE121 elements was updated during the
computation by the code, through a re-meshing operation,
?EDA Publishing/DTIP 2007
ISBN: 978-2-35500-000-3
A.Ballestra, E.Brusa, M.Gh. Munteanu, A.Som?
Experimental characterization of the static behavior of microcantilevers electrostatically actuated
or ?morphing?, applied to the geometrical nonlinear
solution.
Analytical Experimental
N. Voltage
[V]
Displ.
[?m]
Voltage
[V]
Displ.
[?m]
ST1-1 180 0,92 184 1,6
ST1-2 480 1,64 N.A. N.A.
ST1-3 1253 2,69 N.A. N.A.
ST1-4 126 1,64 136 3
ST1-5 323 2,70 N.A. N.A.
ST1-6 86 3,92 crashed crashed
ST1-7 546 6,36 N.A. N.A,
ST1-8 1137 6,88 N.A. N.A,
Table 2: Validation of the analytical model on pull-in.
Since Young modulus of the material was known only
approximately, some measures were performed through a
dynamic response of the microcantilevers [1, 12, 21].
Results showed a certain variability of the values,
therefore a minimum and a maximum value of 150 GPa
and 166 GPa respectively were inputted into the
simulation to investigate the model sensitivity on this
parameter, and results were drawn in Figures 5-7.
4. DISCUSSION
The influence of the geometrical nonlinearity due to the
large displacement of the tip of the tested microcantilevers
is sufficiently high to motivate the implementation of a
nonlinear structural and coupled analysis. As figures 5 and
6 show, the behavior close to pull-in condition becomes
nonlinear and differences with linear solution are
remarkable. Specimen ST1-6 exhibits the same behavior,
but the accidental failure of the specimens did not allow to
reach the pull-in. Value of Young modulus affects the
computation of pull-in, but not significantly like the
thickness. Experiments show that nominal values of E
never fitted the actual response of the structure, but all
results are enclosed in the area delimited by the curves
computed with the two selected values. Results of the
nonlinear model based on SFET element are consistent
with the actual behavior of the specimens, although the
value of pull-in voltage is always predicted with a certain
approximation. The coupled field approach, with
morphing, based on elements PLANE121/183,
overestimated a little bit the actual behavior in above
tests.
Legend
???? Linear (166 GPa) ? Experiments
? ? Non incremental (150 GPa) ? ? ? (166 GPa)
? PLANE121 / 183 (150 GPa) + (166 GPa)
Figure 5: Comparison for specimen ST1-1.
Figure 6: Comparison for specimen ST1-4.
Figure 7: Comparison for specimen ST1-6.
Nevertheless it validated and confirmed the prediction of
the proposed FEM approach [16-18]. Computational
times are comparable for these two FEM solutions.
Authors are currently investigating the possibility to
enhance the performances of the solution algorithms by
?EDA Publishing/DTIP 2007
ISBN: 978-2-35500-000-3
A.Ballestra, E.Brusa, M.Gh. Munteanu, A.Som?
Experimental characterization of the static behavior of microcantilevers electrostatically actuated
means of a discretization based on mixed methods not
only FEM-BEM, but also on the Cell Method.
5. CONCLUSIONS
This study was aimed to validate the numerical
approaches proposed in [14-18] to predict the static
deflection of microbeams electrostatically actuated.
Experiments demonstrated that, even for small values of
aspect ratios described in Table 1, the geometrical
nonlinearity, mainly due to large displacements in
microcantilevers is influent and should be implemented
into the numerical design tools. The non incremental
approach [17, 18] gives results consistent with the
experiments, and compatible with the coupled-field,
nonlinear and iterative solution available in commercial
codes like ANSYS. Computational time is comparable for
the two FEM approaches. Current activity carried by the
authors concerns some additional effects, due to the three
dimensional nature of the electric field, a double clamped
layout and residual stresses, and the discretization of the
dielectric field by Cell Method to enhance the
computational performance.
7. AKNOWLEDGEMENTS
This work was partially funded by the Italian Ministry of
University, under grant PRIN-2005/2005091729, while
microspecimens were built by STMicroelectronics
(Cornaredo, Italy).
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?EDA Publishing/DTIP 2007
ISBN: 978-2-35500-000-3